It remains to consider \(t < \tau _0 = B \log K_n\) . It follows from a union bound and Bernstein's inequality, see Corollary 2.11 in [1]}, that for any constant \(C > 0\) ,
\(\begin{split}&\mathbb {P}\left(\max _{k = 1,\ldots ,K_n} \max _{t = 1,\ldots ,\tau _0-1} \left(\sum _{l = 1}^{t} Y_{nkt}\right)^2 / t > C (\log K_n)^2\right)\\\le & \sum _{k = 1}^{K_n}\sum _{t = 1}^{\tau _0 - 1} \left\lbrace  \mathbb {P}\left(\sum _{l = 1}^{t} Y_{nkt} > \sqrt{C t} \log K_n \right) + \mathbb {P}\left(\sum _{l = 1}^{t} Y_{nkt} < - \sqrt{C t} \log K_n \right) \right\rbrace \\\le & 2 K_n B \log K_n \exp \left( - \sqrt{C} / (\sqrt{2} + c) \log K_n \right)= 2 B K_n^{- \sqrt{C} / (\sqrt{2} + c) - 1} \log K_n.\end{split}\) 
